Speaker
Description
We explore O(2N) scalar theory with fractional Laplacian in d-dimension, √–∇² and its Hamiltonian dynamics which is described by a Schrödinger type equation. In classical limit, this reduces to Hamilton-Jacobi equation, which is widely used to describe holographic renormalization group of multi-trace deformation in dual field theory. This equation is a kind of current conservation equation, where one can define a current j(φ) of a probability P(φ), where φ is the scalar field. Naturally, Gibbs entropy S = –∫ [Dφ] P(φ) log P(φ) can be considered to explore the system. We realize that this Gibbs entropy of the O(2N) scalar theory with fractional Laplacian is matched with free energy of O(N) vector model in finite temperature with chemical potential in d-dimension. The precise map between the stochastic fictitious time t and the inverse temperature β is β = 2t. Therefore, the temperature dependence of the thermal O(N) vector model can be realized as a dynamics of time dependent solution satisfying Schrödinger type equation. This free energy is obtained by putting O(N) vector model in S¹ × R_d, where S¹ is thermal circle with its periodicity β. To get d-dimensional theory, we sum up all possible frequencies on the circle (so called Matsubara frequency summation) which gives d-dimensional thermal partition function. We note that the nontrivial t-dependence appears beyond classical limit.
To take into account quantum effects, we solve the Hamiltonian dynamics by keeping ħ corrections. The chemical potential is mediated by a parameter l and so we call this l-deformation. This is related to the boundary condition of the Schrödinger equation. We also note that the two theories are not equivalent to each other and we see their correspondence in the level of one-loop determinant, i.e. zero point function.
